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Chapter 6: Problem 58

Solve. $$ a^{2}+18 a=0 $$

Short Answer

Expert verified
a = 0 or a = -18

Step by step solution

01

Factor out the common term

Notice that both terms in the equation share a common factor of 'a'. Factor 'a' out of both terms of the equation: a^{2} + 18a = a(a + 18) = 0
02

Set each factor to zero

Using the zero-product property, set each factor equal to zero and solve for 'a': a = 0 a + 18 = 0
03

Solve for 'a'

Solve each equation separately: For a = 0, the solution is already provided: a = 0 For a + 18 = 0, solve for 'a': a = -18

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring
Factoring is a critical concept when solving quadratic equations. It involves expressing a polynomial as the product of its simpler factors. In our given exercise, the equation is \(a^2 + 18a = 0\). We start by identifying the common factors of each term. Here, both terms share the factor \(a\). This allows us to rewrite the equation as \(a(a + 18) = 0\). By factoring out \(a\), we simplify the equation into a product of two factors. This makes it easier to apply the next concept, the zero-product property.
zero-product property
The zero-product property is a fundamental principle in algebra. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is essential when solving equations that have been factored. In our factored equation \(a(a + 18) = 0\), the zero-product property tells us that either \(a = 0\) or \(a + 18 = 0\). By setting each factor equal to zero, we create two simpler equations that are easier to solve for \(a\).
quadratic equation solutions
Solving quadratic equations involves finding the values of the variable that make the equation true. After factoring and applying the zero-product property, we are left with two separate equations: \(a = 0\) and \(a + 18 = 0\). Solving the first equation \(a = 0\) is straightforward. For the second equation \(a + 18 = 0\), we subtract 18 from both sides to get \(a = -18\). These are the solutions to the quadratic equation \(a^2 + 18a = 0\). Every quadratic equation has at most two solutions, as seen here.
common factors
Finding common factors is the initial step in the process of factoring. A common factor is a number or variable that divides each term in the polynomial evenly. In the equation \(a^2 + 18a = 0\), the common factor is \(a\). Recognizing and factoring out the common factor simplifies the equation and makes it manageable. This is why understanding and identifying common factors is an essential part of solving quadratic equations through factoring.

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Most popular questions from this chapter

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